Reading and Plotting Coordinates

Transcription

1 The Coordinate Plane Reteaching Inv. 1 Math Course 3, Investigation 1 Two perpendicular number lines form a coordinate plane. -ais the horizontal number line y-ais the vertical number line origin the point at which the -ais and the y-ais intersect quadrants the four regions that make up a coordinate plane. The quadrants are numbered counterclockwise beginning with the upper right as the first. Second Quadrant Third Quadrant 7 8 y First Quadrant Fourth Quadrant Any point on the coordinate plane can be identified with two numbers. These two numbers are called coordinates of the point. Coordinates are written as a pair of numbers in parentheses, for eample: (3, 7). 7 8 Reading and Plotting Coordinates The first number shows the horizontal direction and distance from the origin. The second number show the vertical direction and distance from the origin. The sign of the number indicates the direction: Positive coordinates are to the right or up. Negative coordinates are to the left or down. ( 4, 2) (3, 1) 2 3 ( 2, 3) y (3, 7) 7 8 Use graph paper for these activities. 1. Draw a coordinate plane. Graph and 2. The vertices of rectangle KLMN are label the following points on the located at coordinates (3, 4), (4, 1), coordinate plane: ( 2, 1), ( 2, 3). Plot the coordinates (4, 7) ( 5, 2) (0, 3) ( 2, 6) ( 4, 1) and find its perimeter and area. 3. the quadrant in which each point lies. P = A = ( 2, 7) (3, 4) ( 4, 8) Saon Math Course 3 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 11

2 Pythagorean Theorem Reteaching Math Course 3, Investigation 2 Inv. 2 Pythagorean Theorem If a triangle is a right triangle, then the sum of the squares of the legs equals the square of the hypotenuse. a 2 + b 2 = c 2 Here is a picture that illustrates the Pythagorean Theorem. The areas of the two smaller squares together equal the area of the largest square. c 2 b 2 a 2 The Pythagorean Theorem applies to all right triangles and only to right triangles. A triangle with sides 3 cm, 4 cm, and 5 cm long is a right triangle because = 5 2. Sketch squares on the sides of these triangles. Calculate the area of each square. Then use that information to find the length of the unknown side. If the side is irrational you may leave the answer in square root form or use a calculator and round the answer to one decimal place cm 8 cm 2. 1 in. 1 in. 22 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saon Math Course 3

3 Classifying Quadrilaterals Reteaching Inv. 3 Math Course 3, Investigation 3 Quadrilateral a four-sided polygon Parallelogram a quadrilateral with two pairs of parallel sides Trapezoid a quadrilateral with one pair of parallel sides Isosceles Trapezoid a trapezoid in which the non-parallel sides are equal in length Trapezium a quadrilateral with no pairs of parallel sides Kite a trapezium that has two pairs of adjacent sides of equal length Rectangle a quadrilateral with four right angles Rhombus a quadrilateral with four equal-length sides Identify each of the following quadrilaterals. If they qualify as more than one type, include each Saon Math Course 3 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 33

4 Drawing Geometric Solids Reteaching Math Course 3, Investigation 4 Inv. 4 Refer to Investigation 4 for descriptions and illustrations of geometric solids. The terms face, edge, and verte refer to specific features of solids. One-Point Perspective Drawing: vanishing point 1. Triangular Prism Rectangular Prism Heagonal Prism A rectangular prism has how many faces, edges, and vertices? 2. each figure. a. b. c. d. Which of the above figures is not a polyhedron? 3. Draw a cone. First draw an ellipse as the base. Then draw a dot for the ape, or peak. Draw segments from the opposite points of the circular base to the ape. 4. Create a one-point perspective drawing for a prism. Begin by drawing a polygon in the foreground. Then pick a location for the vanishing point. Then lightly draw segments from the vertices of the polygon to the vanishing point. Finally, draw corresponding segments parallel to the sides of the polygon in the foreground. 44 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saon Math Course 3

5 Graphing Transformations Reteaching Inv. 5 Math Course 3, Investigation 5 Refer to Investigation 5 for descriptions and illustartions of transformations. Perform the following transformations in the space provided. 1. Draw ABC with A at (4, 5) B at 2. Draw KLM with K at ( 2, 0) L at (2, 1) and C (6, 1). Then draw its (7, 0) and M ( 2, 3). Then draw its reflection across the -ais. Correctly image K L M after a 90 rotation label the corresponding vertices about point K. of A B C. y y 3. Draw PQR with P at ( 6, 7) Q at 4. Draw XYZ with X at ( 2, 4) Y at (0, 7) and R ( 3, 1). Then draw its ( 2, 6) and R (4, 6). Then draw its image P Q R after a translation image X Y Z after a contraction of of (5, 3). scale factor 1 2. y y Saon Math Course 3 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 55

6 Reteaching Inv. 6 Collect, Display, and Interpret Data Math Course 3, Investigation 6 Refer to Investigation 6 to review the collection, display, and interpretation of data. 1. A researcher surveyed a sampling of 8th grade students to determine which type of music they preferred. Students were given five choices: rock, blues, country, pop, and rap. In the space below, display the following data in a bar graph. Rock: 9; Blues: 2; Country: 5; Pop: 5; Rap: The 9th graders all had run a mile at the end of the school year. Below are their times: Mile Times 5:27 12:10 6:45 7:44 5:59 7:31 5:44 7:35 7:14 9:31 6:55 5:20 6:52 7:51 8:28 8:21 9:52 9:11 Display this data in a histogram. Sort the data into one minute intervals. 3. In a poll taken to predict the final results for class president, the following results were collected: Out of 80 students randomly polled: 40 said they would vote for Julia Martinez. 28 said they would vote for Beth Caldwell. 12 were undecided. In the space below, create a table and circle graph with the data. Response Number % 66 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saon Math Course 3

7 Probability Simulation Reteaching Inv. 7 Math Course 3, Investigation 7 Suppose a fruit juice company states on its bottles, One in four get their net bottle of juice free! Just look inside the cap to see if you re a winner. If you decide to purchase a bottle of their juice, what is the probability that you will win a free bottle of juice? The theoretical probability is 1 4. This means that, out of all the juice bottles produced for the contest, one-fourth contain a cap with winner printed on it. Thus, on average, one in four bottles is a winner. However, this does not mean that if you buy four bottles of juice you are guaranteed a winner. To determine the probability of finding at least one winner in four purchases, conduct this simulation: Step One: Place three blue marbles and one yellow marble in a bag. Step Two: For each trial pick and replace four marbles one at a time. Keep track of your results. Make a chart that looks like this: Pick 1 Pick 2 Pick 3 Pick 4 At least 1 winner Trial 1 Trial 2 1. Answer the following questions about the above simulation. a. What did one pick from the bag represent? b. What did each four-pick trial represent? c. What did the picking of a yellow marble represent? d. What did the picking of a blue marble represent? e. Eplain: If one in four bottles contains a winning cap, does buying four bottles guarantee a winning cap? Why or why not? Saon Math Course 3 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 77

8 Reteaching Inv. 8 Scatterplots Math Course 3, Investigation 8 Scatterplots can show if there is a relationship between two sets of data. Below is a chart of the height and weight of a group of middle school boys and a scatterplot of the same data. Height (in.) Weight (lbs) Height (in.) Height and Weight of Middle School Boys Weight (lbs.) Although the points do not lie on a straight line, we see that the points have a somewhat linear relationship. A line of best fit can be drawn on the graph that passes near all of the points. This means that a correlation eists between the height and weight of this group of middle school boys. Quantities can be positively correlated, like the above correlation; negatively correlated; or not correlated. 1. Carlandra wants to determine if there is a correlation between the heights of fathers and sons at a certain age. Make a scatterplot of the data. Are the quantities positively or negatively correlated? Sketch a line of best fit. Fathers heights (in.) Sons heights (in.) Sons heights (in.) Fathers heights (in.) 88 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saon Math Course 3

9 Sampling Methods Reteaching Inv. 9 Math Course 3, Investigation 9 When conducting a study, it is necessary to follow certain procedures to make sure the results are as accurate as possible. Here are some tips: 1. When conducting a study that addresses a large number of people, it is often necessary to study a sample, or portion, of the population. 2. To reduce sampling error the sample should be representative, or very similar to the larger population, large enough, and choosen at random. 3. To select your random sample, you can use: a. random sampling: Assign a number to each person and use a random number generator to select a sample of people. b. stratified sampling: Identify groups within the population and randomly choose a sample within each group. 4. Be careful to gather information in a way that is not biased. For a study involving a survey, choose questions carefully, avoiding bias. Your questions should never sway the persons answering them. Identify a problem with each survey below. 1. A survey was conducted at the high school to determine the amount of time people in Cederville spend watching television. 2. A survey was conducted to determine the support for a new ta measure. 25 names were selected from a list of registered voters to answer the survey. 3. In a district-wide school survey to determine how many hours students study each night, researchers took a random sampling of 7th and 11th graders in all schools. 4. To determine the future of an old shopping mall, the city conducted a survey to determine whether citizens of their town would support renovating the mall. On a Saturday, researchers asked every 10 th shopper the following question: Our mall is in such bad shape. We need to renovate the mall so people would shop here more often. Would you support renovating the mall? Saon Math Course 3 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 99

10 Compound Interest Reteaching Math Course 3, Investigation 10 Inv. 10 When you deposit money into a bank, you often earn interest on your money. The interest is the amount of money the bank pays. The amount of money you deposit is called the principal. The bank may pay you simple interest or compound interest. Simple interest is paid on the principal only. For eample, if you deposit $200 into your account that pays 5% simple interest, you would be paid $10 on your account each year. If you take your money out of the bank after three years, you would have a total of $230. Compound interest is paid on the accumulated interest as well as on the principal. If you deposit $200 in an account with 5% annual percentage rate, the amount of interest you would be paid after each year increases. If the earned interest is left in the account, after three years you would have a total of $ After time, compound interest can amount to quite a lot as this table shows: Number of years Simple Interest Compound Interest $230 $300 $500 $700 $ $ $ $ We can epress periodic compound interest using recursive rules for sequences. For eample, the first term in the sequence is the original deposit of $200. Each succeeding term is the product of 1.05 and the previous term. a 1 = 200 a n = 1.05a n 1 Most calculators can compute compound interest. For eample, find the compound interest on your calculator for year number three using these keys. Does it match the amount in the table above? ( ) y or 1. Find the simple interest and the compound interest on a deposit of $500 with a 7% annual interest rate. Put your results in a table that illustrates interest after 3, 5, 10, 20, 40, and 50 years. 110 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saon Math Course 3

11 Non-Linear Functions Reteaching Inv. 11 Math Course 3, Investigation 11 0 sec sec If a ball is thrown from a high ledge, it will not follow a straight-line path to the ground. The path is curved. This kind of curve is called a parabola. We can write a function that models the height of the ball at any given time. The graph of the function is a parabola. The following is a quadratic function: y = It models the height (y) of the ball at any time (). This quadratic function comes from the study of physics and the force of gravity. When we only consider the force of gravity and ignore other forces such as wind and friction, the height of any falling object on Earth can be described by a quadratic equation. We can chart the quadratic function and graph the points. We see that the graph of the height of the falling ball versus the time and the graph of the function are the same: The graphed functions are non-linear functions; their points do not lie on a straight line. The height of the ball is not subject to a constant rate of change. The ball moves faster as time goes by. Practice 47 ft ft 1 sec 1 12 sec t h 50 y seconds feet ft 68 ft George held a rubber ball 4 ft above the ground and hit it with a paddle straight up at 40 ft/sec. He has studied physics and predicted that considering only gravity, the height (h) in feet of the ball at (t) seconds could be described by this function: h = 16t t + 4 a. Make a table to find the height of the ball at 0, 1 2, 1, 2, and seconds. b. Plot the points (t, h). c. What is the maimum height of the ball? d. When does it reach this height? e. Approimately when does the ball hit the ground? 2 a. t h b. h c. d. e t Saon Math Course 3 Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 121

12 Proof of the Pythagorean Theorem Reteaching Inv. 12 Math Course 3, Investigation 12 When mathematicians want to demonstrate that a particular idea is true, they construct a proof. A proof describes how certain given information leads to a certain conclusion. The Pythagorean Theorem states: a 2 + b 2 = c 2. The following proof for the Pythagorean Theorem is based upon the characteristics of similar triangles. 1. Look at the right triangle. C is a right angle. 2. A c b B a C 2. A segment is drawn from verte C across the triangle to AB so that the segment is perpendicular to AB. Point D divides AB into two short segments. 3. c D A b B a C 3. We can label the distance from B to D as. The distance from D to A is c. 4. c c A D b B a C 4. We now have a right triangle ABC which is divided into two smaller right triangles named BCD and ACD. These three triangles are similar. 5. We can write a proportion that relates the hypotenuse of ABC and BCD to corresponding legs of ABC and ACD: 6. We then write a proportion that relates the hypotenuse of ABC and ACD to corresponding legs of ABC and ACD: ABC BCD ABC BCD hyp leg c a = a hyp leg c a = b(c ) 7. Cross multiply the proportions. The term c appears in both cross products. In the first cross product we see that c equals a 2. This means that we can replace c with a 2 in the second cross product. c 2 a 2 = b 2 or b 2 = c 2 + a 2 8. Now we transform the equation by solving for c 2. c 2 = a 2 + b 2 or a 2 + b 2 = c Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saon Math Course 3